Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery

TL;DR

This paper introduces a reparameterized tensor ring functional decomposition using implicit neural representations, enabling high-frequency detail modeling in multi-dimensional data recovery tasks, with improved training and performance.

Contribution

It proposes a novel reparameterization of TR factors combining learnable tensors and fixed bases, enhancing training dynamics and modeling capacity for continuous data representations.

Findings

Achieves superior performance in image inpainting, denoising, super-resolution, and point cloud recovery.

Demonstrates improved training stability and high-frequency detail modeling.

Provides theoretical analysis and a principled initialization scheme.

Abstract

Tensor Ring (TR) decomposition is a powerful tool for high-order data modeling, but is inherently restricted to discrete forms defined on fixed meshgrids. In this work, we propose a TR functional decomposition for both meshgrid and non-meshgrid data, where factors are parameterized by Implicit Neural Representations (INRs). However, optimizing this continuous framework to capture fine-scale details is intrinsically difficult. Through a frequency-domain analysis, we demonstrate that the spectral structure of TR factors determines the frequency composition of the reconstructed tensor and limits the high-frequency modeling capacity. To mitigate this, we propose a reparameterized TR functional decomposition, in which each TR factor is a structured combination of a learnable latent tensor and a fixed basis. This reparameterization is theoretically shown to improve the training dynamics of TR factor learning. We further derive a principled initialization scheme for the fixed basis and prove the Lipschitz continuity of our proposed model. Extensive experiments on image inpainting, denoising, super-resolution, and point cloud recovery demonstrate that our method achieves consistently superior performance over existing approaches. Code is available at https://github.com/YangyangXu2002/RepTRFD.